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    				| 1. | From differential equation of linear S.H.M obtain an expression for acceleration. Velocity and displacement of a particle performing S.H.M. A sononmeter wire 1 meter long weighing 2 g is in resonsnace with a tunning fork of frequency 300 Hz. Find tension in the sonometer wire. | 
| Answer» The differential equation of linear S.H.M. is `(d^(2)x)/(dt^(2))+(k)/(m)x=0` When `m to` mass of the particle performing S.H.M. `(d^(2)x)/(dt^(2))to` acceleration of particles, `k to` force constant. `:. (d^(2)x)/(dt^(2))+(k)/(m)x=0` `:. "Let", (k)/(m)= omega^(2)` `:. (d^(2)x)/(d^(2))+omega^(2)=0` Acceleration, `a=(d^(2)x)/(dt^(2))` `=-omega^(2)x " "...(i)` This is the expression of acceleration, Where, negative sign indicate that acceleration and the dissplacement are in oppsite direction. Now, `v=(dv)/(dt)` `:. a=(d^(2)x)/(dt^(2))` `=(dv)/(dt)=(dv)/(dx)*(dx)/(dt)=(dv)/(dx)*y` `=v*(dv)/(dx)` `:. v*(dv)/(dx)=-omega^(2)x` Intergrating both sides ` int v dv=- omega^(2) int x dx` `:. (v^(2))/(2)=-(omega^(2)x^(2))/(2)+C" "...(ii)` where, C is integration constant. At extreme position `x=+-A` where, A is Amplitude. `0=-(omega^(2)A^(2))/(2)+C` `C=(omega^(2)A^(2))/(2)` On substituting the value of C in equation (i), `(v^(2))/(2)=-(omega^(2)x^(2))/(2)+(omega^(2)A^(2))/(2)` `v^(2)=omega^(2)(A^(2)-x^(2))` `v=+- omega sqrt(A^(2)-x^(2))` On Substituting the value of C in equation (i), `(v^(2))/(2)=-(omega^(2)x^(2))/(2)+(omega^(2)A^(2))/(2)` `v^(2)=omega^(2)(A^(2)-x^(2))` `v=+-omegasqrt((A^(2)-x^(2))` This is the expression for velocity Now,`(dx)/(dt)=omegasqrt(A^(2)-x^(2)) ( :. v=(dv)/(dt))` `(dx)/(sqrt(A^(2)-x^(2)))= omega dt` Integrating both side `int (dx)/(sqrt(A^(2)-x^(2)))int omega. dt` `sin^(-1)((x)/(A))= omega t+alpha` where, `alpha` is integration constant. `:. (x)/(A)= sin (omegat+alpha)` `x=A sin (omega t+alpha)` This is the expression for displacement, where `alpha` depends upon internal condition. Numerical Given `L=1m,=2g=0.002kg,n=300Hz`. `n=(1)/(2L)sqrt((T)/(m))` `T=4n^(2)L^(2)m` `=3(300)^(2)xx(1)^(2)xx0.002` `T=720N` | |